Proceedings TH2002 Supplement (2003) 43 – 71
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/04043-29 $ 1.50+0.20/0
Proceedings TH2002
Phase transitions in self-gravitating systems
Pierre-Henri Chavanis
Abstract. We discuss the nature of phase transitions in self-gravitating systems. We
show the connexion between the binary star model of Padmanabhan, the thermodynamics of classical point mass stars and the thermodynamics of the self-gravitating
Fermi gas at non-zero temperature. We also briefly discuss the influence of rotation
on the thermodynamical stability of self-gravitating systems.
1 Introduction
Self-gravitating systems exhibit peculiar features such as negative specific heats,
inequivalence of statistical ensembles and phase transitions associated with gravitational collapse. These curious behaviours have been recently re-discovered for
other systems interacting via long-range forces in different fields of physics (nuclear
physics, plasma physics, fracture, Bose-Einstein condensates, atomic clusters, twodimensional turbulence...). The main challenge is represented by the construction
of a thermodynamic treatment of systems with long-range forces and by the analogies and differences among the numerous domains of application. In that context,
self-gravitating systems provide a model of fundamental interest for which ideas of
statistical mechanics and thermodynamics can be tested and developed. The main
domain of application is astrophysics but the general methods can be transposed
in other fields of physics [1].
2 Negative specific heats
Certainly, the most striking property of systems interacting via long-range forces
is the existence of negative specific heats. Astronomers are used to this concept
for a long time [2] but it is only in the 1970 that this problem was specifically
discussed in relation with statistical mechanics [3, 4]. Consider a self-gravitating
system in a steady state. According to the Virial theorem, the kinetic energy K
and the potential energy W are related to each other by the relation
2K + W = 3P V,
(1)
where P is the pressure at the edge of the system of volume V . For an isothermal
system K = 32 N kT . Assuming that the boundary terms can be neglected (this is
not quite correct but this simplifies the argumentation), the total energy is
3
E = K + W = −K = − N kT.
2
(2)
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P.-H. Chavanis
Proceedings TH2002
Therefore, the specific heat is negative,
CV =
dE
3
= − N k < 0.
dT
2
(3)
The above argument is straightforward. However, there exist an equivalently
straightforward argument showing that the specific heat must be positive! Indeed,
for a canonical ensemble in thermal equilibrium, the average energy is given by
E =
Ei exp(−βEi )/
exp(−βEi ).
(4)
i
i
This implies
CV =
dE
= kβ 2 (E 2 − E2 ) > 0.
dT
(5)
This apparent paradox was solved by Thirring and Lynden-Bell who realized that,
for long-range interactions, the statistical ensembles are inequivalent [3, 4]. Therefore, negative specific heats are allowed in MCE (microcanonical ensemble) while
they are forbidden in CE (canonical ensemble).
3 The N-star problem
Consider a system of N stars of mass m in gravitational interaction. The equations
of motion (Newton’s equations) can be cast in a Hamiltonian form
∂H
∂H
dri
dvi
=
=−
,
m
,
dt
∂vi
dt
∂ri
N
Gm2
1
.
mvi2 −
H=
2 i=1
|ri − rj |
i<j
m
(6)
In the microcanonical ensemble, the quantity of fundamental interest is the
density of states
N
g(E) = δ(E − H)
d3 ri d3 vi ,
(7)
i=1
which gives the number of microstates with energy E. The entropy and the temperature are then defined by
S(E) = ln g(E),
1
dS
=
(E).
T
dE
(8)
For N ≥ 3, it is necessary to introduce a small-scale cut-off a and a large-scale cutoff R to make the integral in Eq. (7) finite. For N = 2, the integral is convergent
when a → 0.
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Phase transitions in Self-Gravitating Systems
45
In the canonical ensemble, the quantity of fundamental interest is the partition function
N
Z(β) = e−βH
d3 ri d3 vi .
(9)
i=1
The free energy and the average energy are then given by
F =−
1
ln Z,
β
E = −
∂ ln Z
.
∂β
(10)
Small-scale and large-scale cut-offs are always necessary to make the integral in Eq.
(9) finite (even for N = 2). The distribution of energies for a canonical distribution
at temperature T is given by
P (E) =
1
g(E)e−βE .
Z(β)
(11)
This can also be written
P (E) ∼ eJ(E) ,
J(E) = S(E) − βE.
(12)
The density of state (7) and the partition function (9) cannot be evaluated
analytically in general. In the following, we shall describe two particular situations:
the case N = 2 which provides a toy model of self-gravitating systems (binary star)
and the limit N → +∞ in which a mean-field description is exact.
4 Statistical mechanics of a binary star
To illustrate the nature of phase transitions in self-gravitating systems, we shall
first describe the toy model of Padmanabhan [5]. It consists of N = 2 stars of
mass m in gravitational interaction. We assume that the stars are hard spheres of
radius a/2. They are confined in a spherical box of radius R. For this two-body
system, it is possible to compute the integrals (7) and (9) in a closed form. We
define = Ea/Gm2 , t = T a/Gm2 and µ = R/a. In the microcanonical ensemble,
we have
−1
3
1
−
,
(13)
t() =
1+
for −1 ≤ ≤ −1/µ and
t() =
3[(1 + )2 − µ(1 + µ)2 ] 1
−
(1 + )3 − (1 + µ)3
−1
,
(14)
for ≥ −1/µ. The caloric curve T (E) is represented in Fig. 1 for a cut-off parameter
µ = 30. The minimum energy Emin = −Gm2 /a is obtained when the two particles
are in contact with no thermal motion. For low energies, the particles remain close
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P.-H. Chavanis
Proceedings TH2002
2.8
µ=30
2.4
M
η=βGMm/R
2
1.6
1.2
0.8
CE
0.4
P
MCE
0
−5
−3
−1
1
2
Λ=−ER/GM
3
5
7
Figure 1: Caloric curve for the two-body system in microcanonical and canonical ensembles. The region of negative specific heat present in the microcanonical
ensemble is replaced by a phase transition in the canonical ensemble.
together in average; this corresponds to the condensed phase. For E → Emin , the
temperature behaves as T = 13 (E − Emin ) and the specific heat CV 3. For high
energies, the inter-particle distance is of the order of the box radius R in average;
this corresponds to the gaseous phase. For E → +∞, T = E/2 and CV = 2. For
low and high energies, the specific heat is positive because the system feels the
influence of the cut-offs. For intermediate energies, between points M and P , there
is a region of negative specific heat CV < 0. For Emin E < 0, T ∼ −E which
corresponds to CV −1. In this range of energies, the system is self-bound with
a r R, where r is the typical inter-particle distance. The entropy S(E) is
plotted in Fig. 2; it presents a convex intruder in the region of negative specific
heats. The region of negative specific heats is allowed in MCE because the system
is non-extensive.
In the canonical ensemble, the partition function can be written
µ
e1/tx x2 dx,
(15)
Z(t) = At3
1
with A = 32π 4 R3 m9 G3 . This integral can be evaluated perturbatively for low and
high temperatures. For T → 0, E− Emin = 4T and for T → +∞, E = 3T . The
caloric curve E(T ) is represented in Fig. 1. We see that the region of negative
specific heat allowed in the microcanonical ensemble is replaced by a phase transi-
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Phase transitions in Self-Gravitating Systems
47
µ=30
10
S
0
−10
−20
−5
−2
1
2
Λ=−ER/GM
4
7
Figure 2: Entropy versus energy plot for µ = 30. The curve presents a convex
intruder in the region of negative specific heats.
tion in the canonical ensemble. This phase transition connects the gaseous phase
to the condensed phase at a transition temperature Tt . The transition temperature
behaves with the cut-off parameter as Tt ∼ Gm2 /(3a ln µ) for µ → +∞. The latent heat released at the transition is ∆E = Gm2 /a − Tt . The energy distribution
P (E) is represented in Fig. 3. Close to the transition temperature Tt , it presents
two peaks at E1 and E2 corresponding to the condensed phase and to the gaseous
phase respectively. Below or above Tt the peaks are disymmetric and one of the two
phases appears to be more probable than the other. The dependance of the √
caloric
curve with the cut-off parameter is depicted in Fig. 4. For µ ≤ µCT P = 1 + 3 the
region of negative specific heats in MCE and the plateau in CE disappear. This
corresponds to a canonical tricritical point.
We can also consider the case of two fermions in gravitational interaction.
This is similar to the Bohr atom. The energies of the bound states are
En = −
G2 m5
,
4h̄2 n2
n = 1, 2, ..., ∞
2
(16)
2
2h̄
The energy of the ground state is Emin = − Gm
2aB , where aB = Gm3 is the
gravitational Bohr radius. It plays the role of the hard sphere radius in Padmanabhan’s model. The degeneracy of the level n is gn = 2n2 (the factor 2 accounts
for the spin). If we define the entropy by S(E) = ln gn , then S = − ln(−E) + Cst.
In the microcanonical ensemble, we find that T = −E so that the specific heat
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Proceedings TH2002
300
µ=30
P(ε)=exp(S(ε)−βε)
t=0.08
t=0.09
200
tc=0.0825
100
0
−1
0
2
ε=Ea/Gm
1
Figure 3: Distribution of energy in a canonical ensemble at temperature T . The
function P (E) has a characteristic bimodal structure. The most probable energies
E1 and E2 satisfy J (E) = 0, i.e. S (E) = β. They correspond respectively to the
condensed phase and to the gaseous phase on the microcanonical curve (see Fig.
1 and Eq. (8)). At the transition temperature Tt the two phases have the same
probability. Below or above the transition temperature one of the two phases
emerges as the most probable equilibrium state.
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Phase transitions in Self-Gravitating Systems
49
7
µ=2
6
5
η=βGMm/R
µ=7
µ=µCTP
µ=4.5
4
3
µ=30
2
1
0
µ=1000
−1
0
1
2
Λ=−ER/GM
2
Figure 4: Caloric curve for the two-body system in microcanonical ensemble for
various
√ cut-off radius. The canonical phase transition is suppressed for µ ≤ µCT P =
1 + 3.
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P.-H. Chavanis
Proceedings TH2002
CV = −1 is negative. In the canonical ensemble, for sufficiently low temperatures,
the partition function has the form
Z=
n
max
gn e−βEn =
n=1
n
max
2
2n2 e1/tn ,
(17)
n=1
where t = 2aB T /Gm2 . For T → 0, we find that E − Emin = 3exp(3Gm2 /8aB T ),
which is different from the hard sphere model. The specific heat decreases to zero
exponentially rapidly as T → 0. This behaviour is typical of quantum systems.
Apart from this difference, the hard sphere model and the fermion model share
similar properties.
5 Statistical mechanics of self-gravitating classical particles
We now consider a system of N stars in gravitational interaction, where N 2.
We assume that the system is isolated and that it evolves under the effect of
encounters between stars (“collisional” relaxation). This description is adapted to
globular clusters for which N ∼ 106 . For such systems, the relevant ensemble is the
microcanonical ensemble. For N 1, the density of state can be formally written
in the form
(18)
g(E) = Df eS[f ] δ(E − E[f ])δ(M − M [f ]),
where
S[f ] = −
f ln f d3 rd3 v,
(19)
is the Boltzmann entropy. Physically, the quantity W = eS represents the number of microstates {ri , vi } (specified by the exact position and velocity of the
stars) associated with the macrostate f (r, v) (specified by their smoothed-out distribution). In the thermodynamic limit N → +∞ with Λ = −ER/GM 2 fixed,
the functional integral is dominated by the distribution f which maximizes the
Boltzmann entropy S[f ] at fixed energy
1
1
v2
(20)
E=
f d3 rd3 v +
ρΦd3 r,
2
2
2
and mass
M=
ρd3 r.
(21)
The thermodynamic limit corresponds to N → +∞ with N/R fixed (and
E ∼ 1). This is a very unusual thermodynamic limit due to the nonextensivity of
the system. The condition of extremum δS − βδE − αδM = 0, where β and α are
Lagrange multipliers, leads to Boltzmann’s distribution
f = Ae−β ,
(22)
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Phase transitions in Self-Gravitating Systems
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2
where = v2 + Φ is the stellar energy. The mean-field equation determining the
gravitational potential at equilibrium is then obtained by substituting Eq. (22) in
the Poisson equation
∆Φ = 4πG f d3 v.
(23)
As indicated in Sec. 3, there is no entropy maximum in an infinite domain:
one can increase the entropy of a cluster of stars indefinitely by forming a dense
core and spreading the density of the halo. In addition, the variational problem
leading to the Boltzmann-Poisson equation (22)-(23) has no solution (that could
be a local entropy maximum) since an unbounded isothermal sphere has an infinite
mass. The absence of entropy maximum in an unbounded domain simply means
that a stellar system has the tendency to evaporate under the effect of encounters
between stars. In reality, stellar systems are not allowed to extend to infinity. For
example, globular clusters are subject to the tides of a nearby galaxy so that the
Boltzmann distribution has to be truncated at high energies. The Michie-King
model,
A(e−β − e−βm ) < m ,
f=
(24)
0
≥ m ,
which is a truncated isothermal, can take into account the evaporation of high
energy stars and provides a good description of about 80% of globular clusters.
It can be derived from the Fokker-Planck equation (taking into account the encounters between stars) by imposing that the distribution function vanishes at the
escape energy = m . In that case, the system is not truly static since it gradually
loses stars but we can consider that a globular cluster passes by a succession of
quasi-equilibrium configurations. Instead of working with truncated models, it is
more convenient to confine the system within a box of radius R, the box radius
playing the role of a tidal radius.
The caloric curve β(E) is represented in Fig. 5. For high energies and high
temperatures, we have the perfect gas law E = 32 N kT . For lower values of energy and temperature, the caloric curve forms a spiral. The density contrast
R = ρ(0)/ρ(R) increases from 1 (homogeneous system) to +∞ (singular isothermal
sphere) as we proceed along the spiral. The limit point (Λs , ηs ) = (1/4, 2) corresponds to the density profile ρ = 1/2πGβr2 . Stable solutions in the microcanonical
ensemble correspond to maxima of entropy at fixed mass and energy. There is no
global maximum of entropy for a self-gravitating system confined within a box [6]:
one can increase entropy indefinitely by approaching two stars at infinitely close
distance and releasing the gravitational energy in the halo in the form of thermal
energy. The absence of global entropy maximum reflects the natural tendency of
an isolated stellar system to form binaries. However, the formation of binaries
can take extremely long times, much larger than the age of the universe. Indeed,
equilibrium statistical mechanics tells nothing about the relaxation time. For the
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P.-H. Chavanis
2.5
ηc=2.52
R=32.1
CE
Proceedings TH2002
isothermal collapse
η=βGM/R
MCE
singular
sphere
Λc=0.335
R=709
1.5
gravothermal
catastrophe
0.5
−0.3
−0.1
0.1
0.3
0.5
2
Λ=−ER/GM
0.7
0.9
Figure 5: Caloric curve for classical isothermal spheres. For Λ > Λc or η > ηc , there
is no hydrostatic equilibrium for isothermal spheres and the system undergoes a
gravitational collapse.
timescales contemplated, globular clusters can be considered as metastable equilibrium states corresponding to local entropy maxima. These gaseous metastable
states form the upper branch of Fig. 5 until the point M CE of minimum energy. This corresponds to a density contrast of 709. More concentrated isothermal
spheres are thermodynamically unstable (they are saddle points of entropy). The
probability of transition from a gaseous state to a condensed state (involving a
binary) is extremely small [7, 8]. Therefore, the metastable states are long-lived.
However, because of evaporation, the energy of a globular cluster slowly decreases
with time until it reaches the critical value Ec = −0.335GM 2/R (Antonov energy)
at which the system cannot be in isothermal equilibrium anymore [6]. At that
point, the system undergoes a gravitational collapse called gravothermal catastrophe [9]. The collapse is usually followed with the orbit averaged Fokker-Planck
equation [10]. The evolution proceeds self-similarly with the formation of a high
density core with a shrinking radius and a small mass. The density profile behaves
like ρ ∼ r−α with α 2.2. This core collapse concerns typically 20% of globular
clusters. In theory, the central density becomes infinite in a finite time. In practice, the formation of binaries can release sufficient energy to stop the collapse and
even drive a reexpansion of the system. Then, a series of gravothermal oscillations
should follow.
We now consider a system of self-gravitating particles in contact with a heat
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Phase transitions in Self-Gravitating Systems
53
bath imposing its temperature T . In that case, the proper description is the canonical ensemble [11]. This ensemble may be appropriate to planet, star or galaxy
formation. It is also exact for a system of self-gravitating Brownian particles [12].
The partition function can be written
(25)
Z(β) = Df eJ[f ] δ(M − M [f ]),
where J[f ] = S[f ] − βE is the free energy (multiplied by −β). In the thermodynamic limit N → +∞ with η = βGM/R fixed, statistical equilibrium corresponds
to a configuration that maximizes the free energy J at fixed mass and temperature.
The condition δJ − αδM = 0 again leads to the Boltzmann distribution (22). The
caloric curve E = E(β), where E must be viewed as the average energy, is deduced from Fig. 5 by a rotation of 90◦ . Stable solutions in the canonical ensemble
correspond to maxima of free energy at fixed mass and temperature. In fact, there
is no global maximum of free energy: one can increase the free energy indefinitely
by collapsing all particles at r = 0. Therefore, the natural tendency of a canonical distribution of self-gravitating particles is to form a Dirac peak. There exist,
however, local maxima of free energy. These gaseous metastable states form the
upper branch of Fig. 5 (rotated by 90◦ ) until the point CE of minimum temperature. This corresponds to a density contrast of 32.1. More concentrated isothermal
spheres are unstable in the canonical ensemble (they are saddle points of free
energy). It can be noted that the region of negative specific heats between CE
and M CE is stable in the microcanonical ensemble but unstable in the canonical
GM
, the system cannot be in isothermal
ensemble, as expected. For T < Tc = 2.52R
equilibrium and it undergoes a gravitational collapse. The evolution is usually
followed with the Navier-Stokes equations with an isothermal equation of state
p = ρT [13]. For self-gravitating Brownian particles, the evolution is described
by the Smoluchowski-Poisson system [12, 14]. It is found that the collapse is selfsimilar and that the density profile behaves like ρ ∼ r−α with α = 2. The Dirac
peak is formed in the post-collapse regime after the singularity has arised [14].
The inequivalence of statistical ensembles regarding the formation of binaries (in
MCE) or Dirac peak (in CE) is discussed in [14].
6 Statistical mechanics of self-gravitating fermions
We now consider a system of self-gravitating fermions [15]. Statistical equilibrium
corresponds to a configuration that maximizes the Fermi-Dirac entropy
f
f
f
f
S=−
ln
+ 1−
(26)
ln 1 −
d3 rd3 v,
η0 η0
η0
η0
at fixed mass and energy. This form of entropy can be obtained by a standard
combinatorial analysis preventing two fermions with the same spin to occupy the
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same microcell. The condition of extremum δS − βδE − αδM = 0 leads to the
Fermi-Dirac distribution
η0
f=
,
(β = m/kT ).
(27)
v2
1 + λeβ( 2 +Φ)
In Eq. (27), η0 represents the maximum allowable value of the distribution
function, i.e. f ≤ η0 . The Fermi-Dirac distribution describes either quantum particles (e.g., electrons or neutrons in white dwarfs and neutron stars, massive neutrinos in dark matter models) [16] or collisionless stellar systems experiencing
violent relaxation (in the two-levels approximation) [17, 18]. For a quantum gas,
η0 = (2s + 1)m4 /h3 where s is the spin of the particles. For collisionless stellar
systems, η0 is the maximum value of the initial distribution function. In the nondegenerate limit f η0 , we recover the isothermal distribution (22). In the fully
degenerate limit f η0 , the system is equivalent to a polytrope of index n = 3/2.
A truncated model keeping track of the degeneracy is given by
−βη0 −βη0 m
< m ,
η0 e λ+e−e
−βη0 (28)
f=
0
≥ m ,
where m is the escape energy [19]. In the following, we shall enclose the gas within
a box of radius R and use the complete Fermi-Dirac distribution (27).
When degeneracy is taken into account, the structure
√ of the caloric curve depends on the value of the degeneracy parameter µ = η0 512π 4 G3 M R3 [18]. The
classical limit is recovered for µ → +∞. We see that the inclusion of degeneracy
has the effect of unwinding the spiral of Fig. 5. For large values of the degeneracy
parameter, the caloric curve is represented in Fig. 6. We first discuss the microcanonical ensemble by taking the energy E as control parameter. The stability
of the solutions can be settled by using a turning point criterion [20, 18]. The
solutions on the wingling branch between Λc and Λ∗ (µ) are unstable saddle points
(SP) of entropy; the other solutions are entropy maxima (EM). The solutions on
the upper branch (points A) are non degenerate and have a smooth density profile;
they form the “gaseous phase”. The solutions on the lower branch (points C) have
a “core-halo” structure with a massive degenerate nucleus and a dilute atmosphere;
they form the “condensed phase”. The density profiles of these solutions are shown
in Fig. 7. The degenerate nucleus of the condensed configuration is clearly visible. The entropy versus energy plot is represented in Fig. 8. There is a crossover
at the transition energy Et (µ). This corresponds to a microcanonical first order
phase transition (or gravitational phase transition) connecting the gaseous phase
to the condensed phase. It is accompanied by a discontinuity of temperature and
specific heat (passing from positive to negative values). For µ → +∞ (classical
limit), Et (µ) is rejected to infinity. For E < Et (resp. E > Et ), the gaseous (resp.
condensed) states are metastable as they correspond to local entropy maxima. As
explained previously, these metastable states are long-lived because the probability
of a fluctuation able to trigger the phase transition is extremely weak. Indeed, the
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Phase transitions in Self-Gravitating Systems
55
2.5
µ=10
5
A
LEM
2
Λc=0.335
η=βGM/R
B
1.5
SP
Collapse
GEM
1
Explosion
0.5
Λ*(µ)
LEM
Λt(µ)=−0.28
GEM
C
0
−1
−0.7
−0.4
−0.1
2
Λ=−ER/GM
D
0.2
0.5
Figure 6: Caloric curve of the self-gravitating Fermi gas with a degeneracy parameter µ = 105 . Points A form the “gaseous” phase. They are global entropy
maxima (GEM) for Λ < Λt (µ) and local entropy maxima (LEM), i.e. metastable
states, for Λ > Λt (µ). Points C form the “condensed” phase. They are LEM for
Λ < Λt (µ) and GEM for Λ > Λt (µ). Points B are unstable saddle points (SP) and
contain a “germ”. For Λ ≥ Λc , a gaseous configuration undergoes a gravothermal
catastrophe (collapse) and for Λ ≤ Λ∗ (µ), a condensed configuration undergoes
an explosion. By varying the energy of the system between Λ∗ and Λc , we can
generate an hysteretic cycle between the gaseous and the condensed phase.
P.-H. Chavanis
ln(ρ/<ρ>)
56
16
C
13
B
Proceedings TH2002
µ=10
Λ=0.02
5
10
7
4
A
1
−2
−7
−6
−5
−4
−3
−2
−1
0
ln(r/R)
Figure 7: Density profile of points A, B and C for a degeneracy parameter µ = 105
and an energy Λ = 0.02.
system has to cross the entropic barrier played by the solutions on the intermediate branch (points B). These states contain a small nucleus (with little mass and
energy) which plays the role of a “germ” in the language of phase transitions. In
any case, a phase transition must occur at the critical energies Ec and E∗ at which
the metastable branches disappear. Below Ec , the system undergoes a gravothermal catastrophe but, for self-gravitating fermions, the core ceases to shrink when
it becomes degenerate. Since this collapse is accompanied by a discontinuous jump
of entropy, this is sometimes called a zeroth order phase transition. The resulting
equilibrium state (point D) possesses a small degenerate nucleus which contains a
moderate fraction of the total mass ( 20% for µ = 105 and this fraction decreases
for larger values of µ). The rest of the mass is diluted in a hot envelope held by
the box. In an open system, the halo would be dispersed at infinity so that only
the degenerate core would remain. Inversely, above E∗ the condensed metastable
branch disappears and the system undergoes a discontinuous transition reversed
to the collapse at Ec . This transition is sometimes called an “explosion”, since it
transforms the dense core into a relatively uniform mass distribution. Since the
collapse and the explosion occur at different values of energy, we can generate an
hysteretic cycle by varying the energy between E∗ and Ec .
For smaller values of the degeneracy parameter, the caloric curve is represented in Fig. 9. The curve T (E) is now univalued and the first order phase transition in the microcanonical ensemble is suppressed: all the equilibrium states are
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13
A
Phase transition
C
Λ*(µ)
Λt(µ)
Sη0/M
12
B
C
A
11
Λc
10
−1
−0.7
−0.4
−0.1
2
Λ=−ER/GM
0.2
0.5
Figure 8: Entropy of each phase versus energy for µ = 105 . A microcanonical
first order phase transition is expected at Λt (µ) at which the two stable branches
(solutions A and C) intersect. However, the entropic barrier played by the solution
B probably prevents this phase transition [18, 8].
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P.-H. Chavanis
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3.5
µ=10
3
3
η=βGM/R
2.5
2
GEM
GEM
1.5
1
GEM
GEM
0.5
Λmax(µ)
0
−4
−2
0
2
2
Λ=−ER/GM
4
6
Figure 9: Caloric curve for Fermi-Dirac spheres with µ = 103 . All the solutions are
global entropy maxima. The part of the curve between the extrema of temperature
has negative specific heat.
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Phase transitions in Self-Gravitating Systems
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1
µ=10
gaseous phase
3
0.8
‘‘bump’’
(Λ∼Λc)
κ= R95/R
0.6
0.4
0.2
condensed
phase
0
−4
−2
0
2
2
Λ=−ER/GM
4
6
Figure 10: Evolution of the order parameter κ = R95 /R with the energy (R95 is the
radius containing 95% of the mass and R is the radius of the whole configuration).
The figure shows the transition from the gaseous phase to the condensed phase
as energy is progressively decreased. The clusterization occurs in the region of
negative specific heats.
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Proceedings TH2002
3.5
µ=10
3
3
η=βGM/R
collapse
ηc~2.52
2.5
Λc=0.335
2
LFEM
GFEM
1.5
SP
ηt(µ)=1.06
1
explosion
0.5
C
B
A
GFEM
η*(µ)
LFEM
Λmax(µ)
0
−4
−2
0
2
2
Λ=−ER/GM
4
6
Figure 11: Equilibrium phase diagram for Fermi-Dirac spheres with µ = 103 . In the
canonical ensemble, the system is expected to undergo a first order phase transition
at a transition temperature Tt . However, the local maxima of free energy may be
long-lived (metastable) and the transition will rather take place at Tc and T∗ .
Below Tc , the gaseous states undergo a gravitational collapse (Jeans instability);
above T∗ , the condensed states undergo an explosion. By varying the temperature
of the system between T∗ and Tc , we can generate an hysteretic cycle between the
gaseous and the condensed phase.
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Phase transitions in Self-Gravitating Systems
61
20
15
Jη0/M
C
Phase
transition
10
ηt(µ)
A
η*(µ)
5
0
A
ηc
C
B
1
2
3
η=βGM/R
Figure 12: Free energy of the gaseous and condensed phases versus temperature
for µ = 103 . In the canonical ensemble, a first order phase transition is expected
at Tt at which the two branches intersect.
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Proceedings TH2002
global entropy maxima. For large energies, they are almost homogeneous (gaseous
phase) and for smaller energies they have a “core-halo” structure with a partially
degenerate nucleus and a non-degenerate envelope (mixed phase). As energy is
progressively decreased, the nucleus becomes more and more degenerate and con2/3
tains more and more mass until a minimum energy Emin = −2.36 G2 M 7/3 η0
at which all the mass is in the completely degenerate nucleus of radius R∗ =
−2/3 −1
0.181 η0
G M −1/3 (condensed phase). In that case, the system has the same
structure as a cold white dwarf star. Therefore, depending on the degree of degeneracy and on the value of energy, a wide variety of nuclear concentrations can
be achieved. In Fig. 10, we have plotted the radius containing 95% of the mass
as a function of energy. The quantity κ = R95 /R which measures the degree of
concentration of the system can serve as an order parameter.
We now consider the canonical situation in which the system is in contact with
a heat bath which imposes its temperature T . Considering again the case µ = 103
in Fig. 11, we note that the curve E(T ) is multi-valued. The region of negative
specific heats between ηc and η∗ corresponds to unstable equilibrium states (saddle
points of free energy). We expect therefore a first order phase transition to occur at
a transition temperature Tt (µ) at which the gaseous and condensed states have the
same value of free energy (see Fig. 12). This phase transition should be accompanied by a huge release of latent heat. This may not be physically realizable and the
true collapse will rather occur at Tc . This point of gravitational collapse coincides
with Jeans instability criterion [11]. The outcome of this collapse is the formation
of a fermion ball containing almost all the mass (for sufficiently large values of
µ). Inversely, the condensed phase will undergo an explosion if it is heated above
T∗ . By varying the temperature between T∗ and Tc we can generate an hysteretic
cycle in the canonical ensemble.
7 Maxwell construction and tricritical points
The deformation of the caloric curve when we vary the degeneracy parameter µ is
represented in Fig. 13. A similar diagram would be obtained for a hard sphere gas
(or a soften potential [8]) instead of fermions. In that case, µ would play the role
of the inverse cut-off radius a. For small values of the degeneracy parameter (or
high cut-offs), the caloric curve has a shape very similar to that obtained in the
toy model of Padmanabhan. In particular, there exists a canonical tricritical point
µCT P 82.5 below which the region of negative specific heats in MCE and the
phase transition in CE are suppressed (see Fig. 14). By contrast, for high values of
the degeneracy parameter (or small cut-offs), the diagrams differ fundamentally.
For the N -body system with N 2, the caloric curve winds up and a microcanonical phase transition occurs. This gravitational phase transition appears for
µ > µMT P where µMT P 2600 defines a microcanonical tricritical point (see Fig.
15). The toy model of Padmanabhan does not exhibit this microcanonical phase
transition because, for N = 2, the density of states is always convergent. There-
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Phase transitions in Self-Gravitating Systems
63
3.5
µ=10
3
µ=10
2
η=βGM/R
2.5
2
1.5
µ=10
1
µ=10
µ=10
0.5
0
−1
−0.5
3
4
5
0
0.5
2
Λ=−ER/GM
1
1.5
Figure 13: Caloric curve of self-gravitating fermions for different values of the
degeneracy parameter µ. For µ 1, the spiral makes several rotations before
unwinding [8]. A similar diagram would be obtained for a hard sphere gas.
fore, for N = 2, there is no gravitational collapse at fixed energy: we need at least
a third particle (or more generally a halo of stars) to capture the gravitational
energy released during the formation of a binary.
The temperature of transition in the canonical ensemble can be obtained by
a Maxwell construction as for the familiar Van der Waals gas. The equal area
Maxwell condition A1 = A2 (see Fig. 14) can be expressed as
EC
EA
(β − βt )dE = 0.
(29)
where EA is the energy of the gaseous phase and EB the energy of the condensed
phase at the transition temperature βt−1 . Since dS = βdE, one has
SC − SA − βt (EC − EA ) = 0.
(30)
Introducing the free energy J = S − βE, we verify that Maxwell’s construction is
equivalent to the equality of the free energy of the two phases at the transition,
i.e.,
JA = JC .
(31)
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4
µ= 70
CTP
η=βGM/R
3
µ= 200
ηt
2
µCTP= 82.5
1
0
µ= 1000
−2
0
2
4
2
Λ=−ER/GM
6
8
Figure 14: Enlargement of the caloric curve near the tricritical point in the canonical ensemble. The temperature of transition ηt (µ) is determined by a Maxwell
construction. For µCT P = 82.5, the Maxwell plateau disappears and the E(T )
curve presents an inflexion point at ΛCT P 0.5, ηCT P 3.06. At that point, the
specific heat is infinite and the transition is second order. This diagram is relatively
similar to the liquid/gas transition for an ordinary fluid.
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Phase transitions in Self-Gravitating Systems
65
3
ηc
2.5
ηgas
η=βGM/R
2
MTP
1.5
Λt
1
ηcond
µ=1000
µMTP=2600
µ=6000
0.5
0
µ=2 10
−1
−0.75
−0.5
−0.25
0
0.25
2
Λ=−ER/GM
0.5
0.75
4
1
Figure 15: Same as Fig. 14 near the tricritical point in the microcanonical ensemble
(µMT P = 2600, ΛMT P 0.38, ηMT P 1.68).
We now consider similarly the gravitational phase transition in the microcanonical ensemble (for sufficiently large values of µ). The equal area Maxwell
condition A1 = A2 (see Fig. 15) can be expressed as
βC
βA
(E − Et )dβ = 0.
(32)
−1
−1
where βA
and βB
are the temperatures of the two phases at the transition energy
Et . Since dJ = −Edβ, one has
JC − JA + Et (βC − βA ) = 0.
(33)
Thus, Maxwell’s construction is equivalent to the equality of the entropy of the
two phases at the transition, i.e.,
SA = SC .
(34)
8 Rotating self-gravitating systems
We shall now briefly discuss the thermodynamics of rotating self-gravitating systems. We can take into account the conservation of angular momentum L =
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4
λ=0.3
3
Λc(λ)
λ=0.2
η=βGM/R
λ=0.1
λ=0
2
Gravothermal
catastrophe
1
0
−0.15
−0.025
0.1
2
Λ=−ER/GM
0.225
0.35
Figure 16: Caloric curve of rotating self-gravitating systems for different values of
angular momentum. The gravothermal catastrophe occurs at higher energies in
the microcanonical ensemble and the isothermal collapse at lower temperatures in
the canonical ensemble [21].
f r ∧ vd3 rd3 v by introducing a Lagrange multiplier Ω in the problem of entropy maximization [21]. The deformation of the caloric curve when we increase
the rotation is represented in Fig. 16. It shows that gravitational instability occurs
sooner in MCE and later in CE.
The caloric curve corresponding to self-gravitating fermions is represented in
Figs. 17 and 18 for different values of degeneracy parameter and angular momentum. For high energies, the system is almost homogeneous (gaseous phase) with
an excess of matter against the box at the equator due to the centrifugal force.
For low energies, the system is completely degenerate (condensed phase) and coincides with a distorted polytrope of index n = 3/2. For intermediate energies,
this “fermion spheroid” is surrounded by an isothermal halo (mixed phase) and
the specific heat is negative. As rotation increases, the spheroidal core flattens and
finally develops a cusp at the equator (Fig. 19) [22]. The maximum rotation ΩK is
reached when the centrifugal force and the gravitational force balance each other
(Keplerian limit). For Ω > ΩK , there is mass shedding. For polytropes with index
n > 0.808 (which is the case for the n = 3/2 degenerate core of self-gravitating
fermions), the system remains axisymmetric when rotation increases [23]. This
differs from the case of incompressible bodies (corresponding to n = 0) which un-
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Phase transitions in Self-Gravitating Systems
67
3.8
5
µ=10
λ=0.3
2.8
λ=0.2
η=βGM/R
λ=0.1
λ=0
1.8
Gravothermal
catastrophe stopped
by degeneracy
0.8
D
−0.2
−1
−0.7
−0.4
−0.1
2
Λ=−ER/GM
0.2
0.5
Figure 17: Caloric curve for rotating self-gravitating fermions with a degeneracy
parameter µ = 105 . For E < Ec in a microcanonical description, the system
undergoes a gravothermal catastrophe leading to a rotating fermion ball containing
a small fraction of mass. The rest of the mass is diluted in the halo which would
disperse to infinity in the absence of a confining wall.
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3.5
µ=10
3
3
Rotating
fermion ball
ηc
η=βGM/R
2.5
Isothermal collapse
stopped by degeneracy
2
1.5
λ=0.01
1
λ=0
0.5
0
−4
−2
0
2
2
Λ=−ER/GM
4
6
Figure 18: Caloric curve for rotating self-gravitating fermions with a degeneracy
parameter µ = 103 . For T < Tc in a canonical description, the system undergoes
an isothermal collapse leading to a rotating fermion ball containing a large fraction
of mass.
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Figure 19: Configuration of the rotating self-gravitating Fermi gas at low energies
and high rotations. The system is in a mixed phase with a degenerate core and
an isothermal halo. The core is equivalent to a distorted polytrope with index
n = 3/2. At Ω = ΩK (Keplerian limit), it develops a cusp at the equator. The
thick line marks the level where density is 0.05 the central density.
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dergo a fission instability at high rotations. The model studied by Votyakov et al.
[24], consisting of a Fermi-Dirac distribution in configuration space, enters in that
second category.
9 Conclusion
We have discussed the thermodynamics of self-gravitating systems with different
types of models. The binary star model of Padmanabhan illustrates in a straightforward manner the inequivalence of statistical ensembles for non-extensive systems.
The thermodynamics of N stars in gravitational interaction (considered as classical point masses) exhibits the phenomenon of gravothermal catastrophe which is
a driving mechanism in the evolution of globular clusters. The thermodynamics of
the self-gravitating Fermi gas at non-zero temperature can explain the nature of
phase transitions leading from a gaseous state to a condensed state (fermion ball).
It can have application for white dwarfs, neutron stars and massive neutrinos in
dark matter models. A hard sphere model yields similar results and can be relevant
to planet formation [25]. In addition to these astrophysical applications, the statistical mechanics of self-gravitating systems can serve as a model of fundamental
interest to treat other systems with long-range interactions.
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Pierre-Henri Chavanis
Laboratoire de Physique Quantique
Université Paul Sabatier
118, route de Narbonne
31062 Toulouse, France
71